Plane Sextics with a Type-E
6Singular Point
A l e x D e g t ya r ev
1. Introduction
1.1. The Subject
This paper concludes the series [11;12], where we give a complete deformation clas-sification and compute the fundamental groups of maximizing irreducible plane sextics with an E-type singular point. (With the common abuse of the language, by the fundamental group of a curveB ⊂ P2 we mean the group π
1(P2 \ B)
of its complement.) Here, we consider sexticsB ⊂ P2 satisfying the following
conditions:
(∗) B has simple (i.e., A–D–E) singularities only, B has a distinguished singular point P of type E6, and
B has no singular points of type E7or E8.
(Singular points of type E7and E8are excluded in order to reduce the lists. Sextics
with a type-E8point are considered in [12], and irreducible sextics with a type-E7
point are considered in [11]. Reducible sextics with a type-E7point, as well as the
more involved case of a distinguished D-type point, may appear elsewhere.) Recall that a plane sexticB with simple singularities only is called maximiz-ing if the total Milnor numberµ(B) assumes the maximal possible value 19. It is well known that maximizing sextics are defined over algebraic number fields (as they are related to singularK3-surfaces). Furthermore, such sextics are rigid: two maximizing sextics are equisingular deformation equivalent if and only if they are related by a projective transformation.
Another important class is formed by the so-called sextics of torus type—in other words, those whose equation can be represented in the formf23+ f32 = 0, wheref2andf3are certain homogeneous polynomials of degree 2 and 3,
respec-tively. (This property turns out to be equisingular deformation invariant.) Each sexticB of torus type can be perturbed to Zariski’s famous six-cuspidal sextic [21], which is obtained whenf2andf3as just described are sufficiently generic. Hence,
the groupπ1(P2\ B) factors to the reduced braid group ¯B3 := B3/(σ1σ2)3; in
particular, it is never finite. (The existence of two distinct families of irreducible six-cuspidal sextics, those of and not of torus type, was first stated by Del Pezzo and then proved by Segre; see e.g. [19, p. 407]. Zariski [21] later showed that the two families differ by the fundamental groups.)
Received July 27, 2009. Revision received January 25, 2010.
A representation of the equation ofB in the form f3
2 + f32= 0 is called a torus
structure. The points of intersection of the conic{f2= 0} and cubic {f3 = 0} are
always singular forB; they are called inner (with respect to the given torus struc-ture), whereas the other singular points are called outer. In the listing that follows (see Tables 1 and 2 in Section 2.6), we indicate sextics of torus type by represent-ing their sets of srepresent-ingularities in the form
(inner singularities)⊕ outer singularities.
An exception is the set of singularities E6⊕ A5⊕ 4A2, which is always of torus
type and admits four distinct torus structures.
Formally, the deformation classification of plane sextics with simple singular-ities can be reduced to a purely arithmetical problem (see [6]), and for maxi-mizing sextics this latter problem has been completely solved (see [18; 20]) in the sense that all deformation classes have been enumerated. Unfortunately, this approach—based on the theory ofK3-surfaces and the global Torelli theorem—is not constructive, and very little is known about the geometry of the curves. (Spo-radic examples using explicit equations are scattered in the literature.) Here we use another approach, suggested in [10] and [11]: a plane curveB with a sufficiently deep (with respect to the degree) singularity is reduced to a trigonal curve ¯B in an appropriate Hirzebruch surface. IfB is a maximizing sextic (with a triple point), then ¯B is a maximal trigonal curve; hence it can be studied using Grothendieck’s dessin d’enfants of its functionalj-invariant. In the end, we obtain an explicit geometric description of ¯B and B (rather than equation); among other things, this description is suitable for computing the braid monodromy and hence the funda-mental group of the curves.
1.2. Results
The principal results of this paper are Theorem 1.2.1 (the classification) and The-orems 1.2.2 and 1.2.3 (the computation of the fundamental group).
1.2.1. Theorem. Up to projective transformation (equivalently, up to equisin-gular deformation), there are 93 maximizing plane sextics satisfying condition(∗) realizing 71 combinatorial sets of singularities; of them, 53 sextics (40 sets of sin-gularities) are irreducible (see Table 1) and 40 sextics (32 sets of sinsin-gularities) are reducible (see Table 2).
In Theorem 1.2.1, one set of singularities is common: E6⊕ A9⊕ A4is realized by
one reducible and three irreducible sextics. This theorem is proved in Section 2; for more details, see comments to the tables in Section 2.6.
Among the irreducible sextics in Theorem 1.2.1, twelve (eight sets of singulari-ties) are of torus type. Seven of them (four sets of singularisingulari-ties) are “new” in the sense that they have not been extensively studied before.
1.2.2. Theorem. LetB ⊂ P2be an irreducible maximizing sextic that satisfies
condition(∗) and is not of torus type. If the set of singularities of B is 2E6⊕A4⊕A3
(nos. 4 and 5 in Table 1), thenπ1(P2\B) = SL(2, F5)Z6(see (3.3.3) or (3.5.2)
1.2.3. Theorem. LetB be a sextic as in Theorem 1.2.1, and let B be a proper irreducible perturbation ofB that is not of torus type. Then π1(P2\ B ) = Z6.
Theorems 1.2.2 and 1.2.3 are proved in Section 3 and Section 4.4, respectively. The theorems substantiate my conjecture asserting that the fundamental group of an ir-reducible plane sextic that has simple singularities only and is not of torus type is finite. Recall that the conjecture was originally motivated by certain experimental evidence (which has now been extended) and the fact that the abelianization of the commutant of the fundamental group of an irreducible sextic that is not of torus type is finite (which is a restatement of the proved part of the so-called Oka conjecture; see [5]). At present, the conjecture is essentially settled for sextics with a triple singular point (the case of a D-type point is considered in a forthcoming paper).
In Section 3 we also write down presentations of the fundamental groups of all other sextics as in Theorem 1.2.1, and in Section 4 we consider their perturbations. In particular, we prove the following theorem, computing the groups of all “new” irreducible sextics of torus type.
1.2.4. Theorem. The fundamental group of a sextic with the set of singularities (E6⊕ A11) ⊕ A2,(E6⊕ A8⊕ A2) ⊕ A3, or(E6⊕ A8⊕ A2) ⊕ A2⊕ A1(nos. 12,
13, 18, and 40 in Table 1) is isomorphic to the reduced braid group ¯B3. The groups
of sextics with the set of singularities(E6⊕ A5⊕ 2A2) ⊕ A4(nos. 9 and 41) are
not isomorphic to ¯B3; their presentations are found in Sections 3.3.9 and 3.6(3),
respectively.
1.2.5. Problem. Are the fundamental groups of the two sextics with the set of singularities(E6⊕ A5⊕ 2A2) ⊕ A4(nos. 9 and 41 in Table 1) isomorphic to each
other? (A similar question still stands for the sextics with the set of singularities (2E6⊕ A5) ⊕ A2, nos. 6 and 7 in Table 1; see [4; 17].)
1.2.6. Theorem. LetB be a sextic as in Theorem 1.2.1, and let B be a proper irreducible perturbation ofB that is of torus type. Then, with the following few exceptions:
• a perturbation E6⊕ A5⊕ 4A2→ (a curve of weight 8) (see [9]) and
• a perturbation that can be further perturbed to a curveB with the set of sin-gularities(6A2) ⊕ 4A1(see [4; 9]),
the groupπ1(P2\ B ) is the reduced braid group ¯B3.
(Here, the weight of a sextic is understood in the sense of [5] as the total weight of all its singular points, where the weightw(P ) of a singular point P is defined via w(A3k−1) = k, w(E6) = 2, and w(P ) = 0 otherwise. The fundamental
group of a sextic of weight ≥ 8 is much larger than ¯B3 because it has a larger
Alexander polynomial. In the second exceptional case in the statement, one has π1(P2\ B ) = B4/σ12σ2σ32σ2; see [4; 9].) Theorem 1.2.6 is proved in Section 4.5.
Acknowledgments. I am grateful to E. Artal Bartolo, who helped me identify some of the groups of curves of torus type, thus making the statements more com-plete, and to I. Dolgachev for his enlightening remarks concerning the history of six-cuspidal sextics.
2. The Classification
2.1. The Settings
We recall briefly some of the results of [11] concerning the construction and the classification of plane sextics satisfying(∗). For details on maximal trigonal curves and their skeletons, see [10] or [11]. We denote byk,k > 0, the geometrically ruled rational surface with an exceptional sectionE of self-intersection −k. 2.1.1. Proposition. There is a natural bijectionφ, invariant under equisingu-lar deformations, between Zariski open (in each equisinguequisingu-lar stratum) subsets of the following two sets:
(1) plane sexticsB with a distinguished type-E6singular pointP, and
(2) trigonal curves ¯B ⊂ 4with a distinguished type- ˜A5singular fiber ¯F.
A sexticB is irreducible if and only if so is ¯B = φ(B), and B is maximizing if and only if ¯B is maximal and stable—that is, has no singular fibers of types ˜A∗∗0, ˜A∗1, or ˜A∗2(these fibers are called unstable).
Up to fiberwise equisingular deformation (equivalently, up to automorphism of k), maximal trigonal curves ¯B ⊂ k are classified by their skeletons and type
specifications. The skeleton Sk= Sk¯B ⊂ S2(which is defined as Grothendieck’s
dessin d’enfants of the functionalj-invariant of ¯B) is an embedded connected bi-partite graph with all•-vertices of valency ≤ 3 and all ◦-vertices of valency ≤ 2. The•-vertices of valency ≤ 2 and ◦-vertices of valency 1 are called singular; they correspond to the unstable and type- ˜E singular fibers of ¯B. Besides, each n-gonal region of Sk (i.e., connected component of the complementS2\ Sk) contains a
single singular fiber of ¯B, which is of type ˜An−1( ˜A∗0ifn = 1) or ˜Dn+4. The type
specification is the function choosing, for each singular vertex and each region of Sk, whether the corresponding fiber is of type ˜A or ˜D, ˜E.
The skeleton and the type specification of a maximal curve ¯B ⊂ kare subject to the relation
#•+ #◦(1) + #•(2) = 2(k − t),
wheret is the number of triple singular points of the curve, #∗(n) is the number of
∗-vertices of valency n, ∗ = • or ◦, and #∗is the total number of∗-vertices. Any pair satisfying this relation gives rise to a unique curve.
Under the assumptions of this paper( ¯B has no unstable fibers or fibers of type ˜E7
or ˜E8), all ◦-vertices of Sk are of valency 2 and all its •-vertices are of valency 3
or 1, the latter corresponding to the type- ˜E6 singular fibers of ¯B. Hence the
pre-ceding vertex count can be simplified to
#• = 2(k − t). (2.1.2)
Furthermore, the◦-vertices can be ignored, with the convention that a ◦-vertex is to be understood at the middle of each edge connecting two•-vertices.
To summarize, the proof of Theorem 1.2.1 reduces to the enumeration of all pairs(Sk, type specification), where Sk ⊂ S2 is a connected graph with all
2.2. The Case of Two Type-E6Points
The only maximizing sextic with three type-E6singular points (no. 1 in Table 1) is
well known; see [4; 17]. Assume thatB has two type-E6singular points. Then Sk
has one monovalent•-vertex and, in view of equation (2.1.2), it can be obtained by attaching the fragment•−−• at the center of an edge of a regular 3-graph Sk with two or four vertices (see [7]). All possibilities resulting in a skeleton Sk with a hexagonal region are listed in Figure 1, where Sk is shown in black and the possi-ble position of the insertion is shown in gray. The sextics obtained are nos. 2–8 in Table 1; all curves are irreducible given the existence of a type- ˜E6singular fiber.
1 ¯1
2 1 32
¯3
(a) (b) (c)
Figure 1 Two type-E6points
2.2.1. Remark. For insertion 2 in Figure 1(c), the skeleton Sk has two hexago-nal regions resulting in two sextics (nos. 6 and 7 in Table 1). There are indeed two distinct deformation families of sextics of torus type with the set of singularities (2E6⊕ A5) ⊕ A2; see [4; 17] and Remark 2.6.1.
2.3. The Case of a Hexagon with a Loop
For the rest of this section, assume thatP is the only type-E6singular point ofB.
Then Sk is a regular 3-graph with a distinguished hexagonal region ¯H. Such skele-tons can be enumerated using [2]; however, we choose a more constructive de-scriptive approach.
Combinatorially, there are three possibilities for ¯H : (i) hexagon with a loop (see Figure 2, left);
(ii) hexagon with a double loop (see Figure 5); or (iii) genuine hexagon (see Section 2.5).
Assume that ¯H is a hexagon with a loop (see the shaded area in Figure 2, left). Removing a neighborhood of ¯H from Sk and patching vertices u, v in the figure to a single edge, one obtains another regular 3-graph Sk with at most four vertices
u v u v
(see [7]). Conversely, Sk can be obtained from Sk by inserting a fragment as in Figure 2, left, at the middle of an edge of Sk . The essentially distinct possibilities for the position of the insertion are shown in Figure 3 for irreducible curves and in Figure 4 for reducible curves (a reducibility criterion is found in [10]). To simplify the drawings, we represent the insertion by a gray triangle (as in Figure 2, right).
The resulting sextics are nos. 9–32 in Table 1 and nos. 1 –19 in Table 2.
1 ¯1 2 3 1 2 3 4 ¯4 5 ¯5 6 (a) (b) (c) 1 ¯1 2 ¯2 3 4 1 2 ¯2 3 1 ¯1 2 (d) (e) (f )
Figure 3 A hexagon with a loop: Irreducible curves
1 2 ¯2 3 1 3 2 (a) (b) 1 2 (c) (d) (e) (f )
Figure 4 A hexagon with a loop: Reducible curves
2.3.1. Remark. The curves in pairs nos. 27, 28, nos. 31, 32, and nos. 15 , 16 in the tables differ by their type specifications: the type- ˜D5fiber can be chosen either
inside one of the “free” loops of Sk shown in the figure or inside the inner loop of the hexagon. We assume that the latter possibility corresponds to curves nos. 28, 32 in Table 1 and no. 16 in Table 2.
2.4. The Case of a Hexagon with a Double Loop
Now assume that the distinguished hexagon ¯H looks like the outer region in Fig-ure 5, left. Each of the remaining fragmentsA, B of Sk has an odd number of
vertices, and the total number of remaining vertices is at most four. Hence, we can assume thatA has one vertex and B has at most three vertices. Then A is a single loop and the graph can be redrawn as shown in Figure 5, right, where ¯H is repre-sented by the shaded area. In other words, Sk can be obtained from a regular 3-graph Sk with two or four vertices and with a loop (see [7]) by replacing a loop with the fragment shown in Figure 5, right. The five possibilities are listed in Figure 6; the resulting sextics are nos. 33–38 in Table 1. (Using [10], one can easily show that the existence of a fragment as in Figure 5, right implies that the curve is irreducible.)
B
A B v
Figure 5 A hexagon with a double loop
2.4.1. Remark. The skeleton in Figure 6(f ) has a symmetry interchanging its two monogons and two pentagons. For this reason, unlike the case described in Remark 2.3.1, items nos. 37 and 38 are realized by one deformation family each.
(b) (c)
(d) (e) (f )
Figure 6 A hexagon with a double loop: The five skeletons
2.5. The Case of a Genuine Hexagon
Finally, assume that ¯H is a genuine hexagon (i.e., all six vertices in the bound-ary∂ ¯H are pairwise distinct). In other words, ∂ ¯H is the equator in S2, and Sk is
obtained from∂ ¯H by completing it to a regular 3-graph by inserting at most two vertices and connecting edges into one of the two hemispheres. All possibilities are listed in Figure 7 for irreducible curves and in Figure 8 for reducible curves; the resulting sextics are nos. 39–42 in Table 1 and nos. 20 –38 in Table 2.
(a) (b) (c) (d)
Figure 7 A genuine hexagon: Irreducible curves
(a) (b) (c) (d)
(e) (f ) (g) (h)
(i) ( j) (k) (l)
(m) (n) (o)
(p) (q)
2.6. The List
The results of the classification are collected in Table 1 for irreducible curves and in Table 2 for reducible curves, where we list the combinatorial types of singular-ities and references to the corresponding figures. (For sextics of torus type, the inner singularities are also indicated; the exception is no. 39 in Table 1, which admits four distinct torus structures.) Equal superscripts precede combinatorial types shared by several items in the tables. (The set of singularities E6⊕ A9⊕ A4
prefixed with10 is common for both tables.) The Count column lists the
num-bers(nr,nc) of real curves and pairs of complex conjugate curves. The last two columns refer to the computation of the fundamental group and indicate the pa-rameters used (explained in the relevant sections). In addition, the curves with nonabelian fundamental groupπ1:= π1(P2\ B) are prefixed with one of the
fol-lowing symbols.
• ∗: The groupπ1is not abelian.
• ?: The groupπ1is not known to be abelian.
• ∗∗: For curves of torus type,π1= ¯B3.
2.6.1. Remark. For pairs nos. 4, 5 and nos. 6, 7, one can ask if the two curves re-main nonequivalent if a permutation of the two type-E6points is allowed. For nos. 6
and 7, there still are two distinct equisingular deformation families (see [4; 17]); for nos. 4 and 5, the two curves become equivalent (see [18]). (Alternatively, if nos. 4 and 5 were not equivalent, then each of the curves would have a symme-try interchanging its two type-E6 points. Because the curve is maximizing, the
symmetry would necessarily be stable—contradicting [7].)
2.6.2. Remark. The sets of singularities nos. 3 and 8 with(nr,nc) = (0, 1) can be realized by real curves; see [18]. However, with respect to this real structure, the two type-E6points must be complex conjugate.
3. The Computation
3.1. Preliminaries and Notation
To compute the fundamental groups, we use Zariski–van Kampen’s method [14], applying it to the ruling of4. The principal steps of the computation are outlined
here; for more details, see [11; 12].
Fix a maximizing sexticB satisfying (∗) and let ¯B be the maximal trigonal curve given by Proposition 2.1.1. For the fiber at infinityF∞we take the distin-guished type- ˜A5fiber ¯F of ¯B, and for the reference fiber F we take the fiber over
an appropriate vertexv of the skeleton Sk of ¯B in the boundary ∂ ¯H of the hexag-onal region ¯H containing ¯F. Choose a marking at v (see [10]) so that the edges e2
ande3 atv belong to the boundary ∂ ¯H, and let {α1,α2,α3} be a canonical basis
inF defined by this marking; see [10] or Figure 9. (The precise choice of the vertexv and the marking is explained below on a case-by-case basis.) Denote ρ = α1α2α3.
Table 1 Maximal Sets of Singularities with a Type-E6Point
Represented by Irreducible Sextics
# Set of singularities Figure Count π1 Parameters
∗∗1 (3E 6) ⊕ A1 (1, 0) see [4] ∗∗2 (2E 6⊕ 2A2) ⊕ A3 1(a) (1, 0) see [4] 3 2E6⊕ A7 1(b)-1 (0, 1) 3.3 (–, –, 1, –) ∗4 12E 6⊕ A4⊕ A3 1(b)-2 (1, 0) 3.5 l = 4 ∗5 12E 6⊕ A4⊕ A3 1(c)-1 (1, 0) 3.3 (4, 5, –, –) ∗∗6 2(2E 6⊕ A5) ⊕ A2 1(c)-2 (1, 0) see [4] ∗∗7 2(2E 6⊕ A5) ⊕ A2 1(c)-2 (1, 0) see [4] 8 2E6⊕ A6⊕ A1 1(c)-3 (0, 1) 3.2 ∗∗9 3(E 6⊕ A5⊕ 2A2) ⊕ A4 3(a) (1, 0) 3.3.9 (6, 5, 3, –) 10 E6⊕ A13 3(b)-1 (0, 1) 3.3 (–, –, 1, –) 11 4E 6⊕ A10⊕ A3 3(b)-2 (1, 0) 3.3.4 ∗12 (E 6⊕ A11) ⊕ A2 3(b)-3 (1, 0) 3.3.7 ∗13 (E 6⊕ A8⊕ A2) ⊕ A3 3(c)-1 (1, 0) 3.3.10 (9, 4, 3, –) 14 5E 6⊕ A7⊕ A4⊕ A2 3(c)-2 (1, 0) 3.3 (5, 8, 3, –) 15 6E 6⊕ A5⊕ 2A4 3(c)-3 (1, 0) 3.3 (5, 5, 6, –) 16 7E 6⊕ A8⊕ A4⊕ A1 3(c)-4 (0, 1) 3.3 (–, –, –, 1) 17 8E 6⊕ A10⊕ A2⊕ A1 3(c)-5 (0, 1) 3.3 (–, –,1, –) ∗18 9(E 6⊕ A8⊕ A2) ⊕ A2⊕ A1 3(c)-6 (1, 0) 3.3.10 (9, 3, –, 3) 19 E6⊕ A7⊕ A6 3(d)-1 (0, 1) 3.3 (–, –, –, 1) 20 10E 6⊕ A9⊕ A4 3(d)-2 (0, 1) 3.3 (–, –,1, –) 21 E6⊕ A6⊕ A4⊕ A3 3(d)-3 (1, 0) 3.3 (4, 7, –, –) 22 5E 6⊕ A7⊕ A4⊕ A2 3(d)-4 (1, 0) 3.3 (3, 8, –, 5) 23 4E 6⊕ A10⊕ A3 3(e)-1 (1, 0) 3.3 (–, –, –, 1) 24 E6⊕ A12⊕ A1 3(e)-2 (0, 1) 3.3 (–, –,1, –) 25 8E 6⊕ A10⊕ A2⊕ A1 3(e)-3 (1, 0) 3.3 (11, 3, –, 2) 26 E6⊕ D13 3(f )-1 (1, 0) 3.3 (–, –,1, –) 27 11E 6⊕ D5⊕ A8 3(f )-1 (0, 1) 3.3 (–, –,1, –) 28 11E 6⊕ D5⊕ A8 3(f )-1 (1, 0) 3.3.6 29 E6⊕ D11⊕ A2 3(f )-2 (1, 0) 3.3 (–, –, –, 1) 30 E6⊕ D7⊕ A6 3(f )-2 (1, 0) 3.3 (–, –, –, 1) 31 12E 6⊕ D5⊕ A6⊕ A2 3(f )-2 (1, 0) 3.3 (7, 3, –, –) 32 12E 6⊕ D5⊕ A6⊕ A2 3(f )-2 (1, 0) 3.3.5 33 10E 6⊕ A9⊕ A4 6(b) (1, 0) 3.5 l = 10 34 13E 6⊕ A6⊕ A4⊕ A2⊕ A1 6(c) (1, 0) 3.5 l = 7 35 6E 6⊕ A5⊕ 2A4 6(d) (1, 0) 3.5 l = 6 36 7E 6⊕ A8⊕ A4⊕ A1 6(e) (1, 0) 3.5 l = 9 37 E6⊕ D9⊕ A4 6(f ) (1, 0) 3.5 38 E6⊕ D5⊕ 2A4 6(f ) (1, 0) 3.5 l = 5 ∗∗39 E 6⊕ A5⊕ 4A2 7(a) (1, 0) 3.6(1) (3, 3, 6, 3, 3, –) ∗40 9(E 6⊕ A8⊕ A2) ⊕ A2⊕ A1 7(b) (0, 1) 3.6(2) (9, –, 3, 3, –, 2) ∗∗41 3(E 6⊕ A5⊕ 2A2) ⊕ A4 7(c) (1, 0) 3.6(3) (6, –, 5, 3, 3, –) 42 13E 6⊕ A6⊕ A4⊕ A2⊕ A1 7(d) (0, 1) 3.6 (7, –, 3, 5, 2, –)
Table 2 Maximal Sets of Singularities with a Type-E6Point
Represented by Reducible Sextics
# Set of singularities Figure Count π1 Parameters
∗∗1 14(E 6⊕ 2A5) ⊕ A3 4(a)-1 (1, 0) 3.4(1) (6, 4, 6, –) 2 15E 6⊕ A7⊕ A5⊕ A1 4(a)-2 (0, 1) 3.4 (–, –, 2, –) ∗3 16E 6⊕ A7⊕ A3⊕ A2⊕ A1 4(a)-3 (1, 0) 3.4(2) (4, 8, –, 3) 4 17E 6⊕ A6⊕ A5⊕ 2A1 4(b)-1 (1, 0) 3.4 (–, –, 2, –) 5 E6⊕ A6⊕ 2A3⊕ A1 4(b)-2 (1, 0) 3.4 (7, 4, 4, –) ∗6 18E 6⊕ A5⊕ A4⊕ A3⊕ A1 4(b)-3 (1, 0) 3.4(3) (5, 6, 4, –) 7 10E 6⊕ A9⊕ A4 4(c)-1 (1, 0) 3.4 (–, –, –, 1) 8 19E 6⊕ A9⊕ A3⊕ A1 4(c)-2 (1, 0) 3.4 (4,10, –, 2) 9 E6⊕ D9⊕ A3⊕ A1 4(d) (1, 0) 3.4 (–, –, 2, –) 10 E6⊕ D8⊕ A4⊕ A1 4(d) (1, 0) 3.4 (–, –, 2, –) 11 E6⊕ D6⊕ A4⊕ A3 4(d) (1, 0) 3.4 (5, 4, –, –) 12 E6⊕ D5⊕ A4⊕ A3⊕ A1 4(d) (1, 0) 3.4.3 (–, –, 2, –) 13 E6⊕ D10⊕ A3 4(e) (1, 0) 3.4 (–, –, –, 1) 14 E6⊕ D8⊕ A5 4(e) (1, 0) 3.4 (–, –, –, 1) ∗15 20E 6⊕ D5⊕ A5⊕ A3 4(e) (1, 0) 3.4(4) (4, 6, –, –) ∗16 20E 6⊕ D5⊕ A5⊕ A3 4(e) (1, 0) 3.4.3 (4, 6, –, 1) 17 E6⊕ D7⊕ D6 4(f ) (1, 0) 3.4.2 18 E6⊕ D7⊕ D5⊕ A1 4(f ) (1, 0) 3.4.3 (–, 2, –, –) 19 E6⊕ D6⊕ D5⊕ A2 4(f ) (1, 0) 3.4.3 (3, –, –, –) ∗∗20 (E 6⊕ A5⊕ 2A2) ⊕ A3⊕ A1 8(a) (1, 0) 3.7(1) (2, 6, 3, 4, 3, –) ?21 E 6⊕ A4⊕ 2A3⊕ A2⊕ A1 8(b) (1, 0) 3.7(2) (2, 5, 4, 3, 4, –) ∗22 E 6⊕ A7⊕ A3⊕ 3A1 8(c) (1, 0) 3.10(1) (2, 8, 2, –, 4, –) ∗23 E 6⊕ A5⊕ 2A3⊕ 2A1 8(d) (1, 0) 3.10(2) (2, 4, 6, 4, –, –) ∗∗24 (E 6⊕ 2A5) ⊕ 3A1 8(e) (1, 0) 3.10(3) (2, 6, 6, 2, –, –) 25 19E 6⊕ A9⊕ A3⊕ A1 8(f ) (1, 0) 3.8 (10, –, 4, 2, –, –) 26 21E 6⊕ A11⊕ 2A1 8(g) (1, 0) 3.8 (12, –, 2, –, 2, –) 27 15E 6⊕ A7⊕ A5⊕ A1 8(h) (0, 1) 3.8 (8, –, 2, –, 6, –) ∗∗28 21(E 6⊕ A11) ⊕ 2A1 8(i) (1, 0) 3.8 (12, –, 2, –, –, 2) ∗∗29 14(E 6⊕ 2A5) ⊕ A3 8( j) (1, 0) 3.8 (6, –, 4, 6, –, –) ∗30 18E 6⊕ A5⊕ A4⊕ A3⊕ A1 8(k) (1, 0) 3.7(3) (6, 5, 4, 2, –, –) 31 17E 6⊕ A6⊕ A5⊕ 2A1 8(l) (1, 0) 3.7 (6, 7, 2, –, 2, –) 32 E6⊕ A7⊕ A4⊕ 2A1 8(m) (1, 0) 3.7 (8, 5, 2, –, –, 2) ∗33 16E 6⊕ A7⊕ A3⊕ A2⊕ A1 8(n) (1, 0) 3.7(4) (8, 3, –, 4, 2, –) ∗34 E 6⊕ A9⊕ A2⊕ 2A1 8(o) (1, 0) 3.7(5) (10, 3, –, 2, –, 2) 35 E6⊕ D8⊕ A3⊕ 2A1 8(p) (1, 0) 3.9 (–, 2, –, 4, 2, –) ∗36 E 6⊕ D6⊕ 2A3⊕ A1 8(p) (1, 0) 3.10(4) (2, 4, 4, –, –, –) ∗37 E 6⊕ D10⊕ 3A1 8(q) (1, 0) 3.10(5) (2, –, 2, –, 2, –) 38 E6⊕ D6⊕ A5⊕ 2A1 8(q) (1, 0) 3.9 (–, 6, 2, –, 2, –)
According to [11], the generatorsα1,α2,α3are subject to the so-called relation
at infinity:
ρ4= (α
s(v) α1
α2
α3
Figure 9 A canonical basis{α1,α2,α3}
LetF1,..., Fr be the singular fibers of ¯B other than ¯F. Dragging F about Fj and
keeping the base point in an appropriate section, one obtains an automorphism mj∈ B3⊂ Autα1,α2,α3, called the braid monodromy about Fj. In this notation,
the groupπ1:= π1(P2\ B) has a presentation of the form
π1= α1,α2,α3| mj = id, j = 1, ..., r, and (3.1.1), (3.1.2)
where each braid relationmj = id, j = 1, ..., r, should be understood as the triple of relationsmj(αi) = αi,i = 1, 2, 3. Furthermore, in the presence of the relation at infinity, (any) one of the braid relationsmj = id, j = 1, ..., r, can be omitted. The braid monodromiesmj are computed using [10]; all necessary details are ex-plained in this section.
Throughout the paper, all finite groups/quotients are analyzed using GAP [13]. Most infinite groups are handled by means of the following obvious lemma, which we state here for future reference.
3.1.3. Lemma. LetG be a group, and let a ∈ G be a central element whose projection to the abelianizationG/[G, G] has infinite order. Then the projection G → G/a restricts to an isomorphism [G, G] = [G/a, G/a].
Given two elementsα, β of a group and a nonnegative integer m, we introduce the notation
{α, β}m= (αβ)
k(βα)−k if m = 2k is even,
((αβ)kα)((βα)kβ)−1 if m = 2k + 1 is odd.
The relation{α, β}m = 1 is equivalent to σm= id, where σ is the generator of the braid groupB2acting on the free groupα, β. Hence,
{α, β}m= {α, β}n= 1 is equivalent to {α, β}g.c.d.(m,n)= 1. (3.1.4)
For the small values ofm, the relation {α, β}m= 1 takes the following form:
• m = 0, tautology;
• m = 1, the identification α = β;
• m = 2, the commutativity relation [α, β] = 1; • m = 3, the braid relation αβα = βαβ.
3.2. Two Type-E6Singular Points
It suffices to consider the set of singularities 2E6⊕ A6⊕ A1(no. 8 in Table 1)
only. The fundamental groups of all sextics of torus type with two type-E6
singu-lar points are found in [4], and the remaining curves that are not of torus type are covered in Sections 3.3 and 3.5.
Thus, consider the skeleton Sk given by Figure 1(c)-3 (i.e., insertion 3 in Fig-ure 1(c)). Letu be the monovalent •-vertex of Sk, and let v be the trivalent •-vertex adjacent tou. Mark v so that [u, v] is the edge e2atv. Then, in addition to
rela-tion (3.1.1), the groupπ1has the relations
{α2α3α−12 ,α1}7= {α1α2α3α−12 α−11 ,α2}2 = 1, ρα2ρ−1= α2α3α−12 ,
obtained from the heptagon, the bigon, and the monovalent•-vertex u of Sk, re-spectively. The resulting group is abelian.
3.3. Hexagon with a Loop: Irreducible Curves
Assume that ¯H is a hexagon with a loop (see Section 2.3 and Figure 10), and take forv the vertex shown in the figure.
w
v r
s t
w
w
Figure 10 A hexagon with a loop: The regions
The inner loop of ¯H (the monogonal region containing w in its boundary) gives the relation
α1= α−13 α2α3. (3.3.1)
Extend Sk to a dessin (see [10]), and consider the×-verticesr, s, t, and w shown in Figure 10. (We do not assert that all these vertices are distinct.) Assume that they are at the centers ofl-, m-, n-, and k-gonal regions of Sk, respectively. Then the braid relations about the singular fibers of ¯B over these vertices are
r: {α1,α2}l = 1, s: {α1,α2α3α−12 }m= 1, t: {α2,ρα3ρ−1}n= 1, w : {α−1 2 α1α2,ρα3ρ−1}k= 1 (3.3.2)
if we assume that the fibers are of type ˜A. For a ˜D-type fiber, we omit the corre-sponding relation in (3.3.2) and indicate this omission by a “–” in the parameter list. (Sometimes, we also omit a relation just because it is not necessary to prove thatπ1is abelian.) Using the values of(l, m, n, k) shown in Table 1, one concludes
that the groups of most sextics that are not of torus type are abelian. The few ex-ceptional cases are treated separately in what follows.
The same arguments apply to the sets of singularities 2E6⊕A7(no. 3 in Table 1)
and 2E6⊕ A4⊕ A3(no. 5 in Table 1) because the corresponding skeletons can
be represented as shown in Figure 11(a)-1, ¯1 and Figure 11(a)-2, respectively. The former fundamental group is abelian. The latter is the order-720 group given by
π1= α1,α2,α3| (3.1.1), (3.3.1), (3.3.2) (3.3.3)
with(l, m, n, k) = (4, 5, –, –); it splits into the semidirect product SL(2, F5) Z6.
(Recall that the braid relation about the remaining type- ˜E6 singular fiber can be
ignored.) An alternative presentation of this group is given by (3.5.2).
1 ¯1
2 3
(a) (b)
Figure 11
3.3.4. The Set of Singularities E6⊕ A10⊕ A3 (no. 11 in Table 1). In
addi-tion to (3.1.1), (3.3.1), and (3.3.2) with(l, m, n, k) = (4,11, –, –), one also has the relation
(α1α2α3α−12 )α1(α1α2α3α−12 )−1= (α−12 α1α2)α3(α−12 α1α2)−1
from the lower left loop in Figure 3(b). The resulting group is abelian.
Alternatively, choosing a canonical basis{α1,α2,α3} in the fiber over the upper
left•-vertex in Figure 3(b), one obtains the relations α2 = α3(from the upper left
loop) andα−12 α1α2 = ρ−1α2ρ (from the lower left loop). In view of the former,
the latter simplifies to the braid relationα1α2α1= α2α1α2; that is, {α1,α2}3= 1.
On the other hand, the 11-gonal outer region of the skeleton gives the relation {α1,α2}11= 1. Hence {α1,α2}1= 1 (see (3.1.4)) and the group is cyclic.
3.3.5. The Set of Singularities E6⊕ D5⊕ A6⊕ A2(no. 32 in Table 1). In
this case we assume that the type- ˜D5singular fiber is chosen inside the inner loop
of the insertion; see Remark 2.3.1. Hence the group has no relation (3.3.1). How-ever, relation (3.1.1) and relations (3.3.2) with(l, m, n, k) = (7, 3, –, 1) suffice to show that the group is abelian.
3.3.6. The Set of Singularities E6⊕ D5⊕ A8(no. 28 in Table 1). As before,
Hence the group has no relation (3.3.1). Still, it has relations (3.1.1) and (3.3.2) with(l, m, n, k) = (9, –,1, –) and, in addition, the relation
α2α3α−12 = (α1α2α3)α2(α1α2α3)−1
resulting from the left loop in Figure 3(f ). These relations suffice to show that the group is abelian.
3.3.7. The Set of Singularities(E6⊕ A11) ⊕ A2(no. 12 in Table 1). As
ex-plained in Section 3.3.4, the groupπ1is a quotient of the braid groupB3(for the
braid relations, only the two left loops in Figure 3(b) are used). Therefore,π1= ¯B3
by the following simple lemma (see e.g. [8, Lemma 3.6.1]).
3.3.8. Lemma. LetB be an irreducible plane sextic of torus type. Then any epi-morphismB3 π1(P2\ B) factors through an isomorphism ¯B3= π1(P2\ B).
3.3.9. The Set of Singularities(E6⊕A5⊕2A2)⊕A4(no. 9 in Table 1). The
groupπ1has presentation (3.3.3) with(l, m, n, k) = (6, 5, 3, –). Using GAP [13],
one can see that the quotientπ1/α15 is a perfect group of order 7680 whereas
¯B3/σ15= A5has order 60. Hence the natural epimorphismπ1 ¯B3is proper.
3.3.10. Other Curves of Torus Type. The remaining sextics of torus type ap-pearing in this section are nos. 13 and 18 in Table 1. The groupπ1has
presenta-tion (3.3.3) with the values of the parameters(l, m, n, k) given in the table. Both groups factor to ¯B3. Entering the presentations into GAP [13] and then simplifying
them via
P := PresentationNormalClosure(g, Subgroup(g, [g.1/g.2])); SimplifyPresentation(P);
one finds that the commutant [π1,π1] is a free group on two generators. Since all
groups involved are residually finite and hence Hopfian, it follows that both epi-morphismsπ1 ¯B3 are isomorphisms. (This approach was suggested to me by
E. Artal Bartolo.)
3.4. Hexagon with a Loop: Reducible Curves
Choose a basis{α1,α2,α3} as in Section 3.3. Then (3.1.1) and (3.3.1) imply that
π1/[π1,π1] = Z and that the projection π1 → Z is given by α1,α2 → 1 and
α3→ −5. In particular, it follows that the sextic splits into an irreducible quintic
and a line.
In addition to (3.1.1) and (3.3.1), consider the relations r: {α1,α2}l = 1, s: {α1,α2α3α−12 }m= 1, t: {α2,ρα3ρ−1}n= 1, w : {α−1 2 α1α2,ρ−1α2ρ}k = 1 (3.4.1)
arising from the×-verticesr, s, t, w in Figure 10 (assuming that the fibers over these vertices are of type ˜A; if a fiber is of type ˜D, the corresponding relation is omitted).
In order to analyze the group using GAP [13], observe that relation (3.1.1) im-plies [α1,(α2α3)3]= 1; then, in view of (3.3.1), the element (α2α3)3commutes
withα3and hence withα2. Thus (α2α3)3∈ π1is a central element, and its
pro-jection to the abelianization ofπ1is the element−12 of infinite order. Because of
Lemma 3.1.3, it suffices to study the commutant of the quotientπ1/(α2α3)3. The
abelianization of the latter quotient isZ12.
The sets of parameters(l, m, n, k) used in the calculation are listed in Table 2. The curves with the following sets of singularities have nonabelian groups. (1) (E6⊕ 2A5) ⊕ A3 (no. 1 ): the curve is of torus type, so ord[π1,π1] = ∞.
Note thatπ1/α12 = GL(2, F3) has order 48; in particular, π1is not isomorphic
to any braid groupBn.
(2) E6⊕ A7⊕ A3⊕ A2⊕ A1(no. 3 ): one has π1= SL(2, F7) × Z.
(3) E6⊕ A5⊕ A4⊕ A3⊕ A1(no. 6 ): one has π1= SL(2, F5) Z.
(4) E6⊕ D5⊕ A5⊕ A3(no. 15 ): one has π1= ((Z3× Z3) Z3) Z.
(In items (2), (3), and (4), the centralizer of [π1,π1] projects to a subgroup of
index 1, 2, and 4, respectively, in the abelianization. In the former case, it follows that the product is direct.) The group in item (2) was first computed in [1]. 3.4.2. The Set of Singularities E6⊕ D7⊕ D6(no. 17 in Table 2). The curve
has no ˜A-type singular fibers outside the insertion, and we replace (3.4.1) with the relations
(ρα1α2α1)α2(ρα1α2α1)−1= α1, (ρα1α2)α1(ρα1α2)−1= α2
resulting from the type- ˜D7singular fiber overr. The resulting group is abelian.
3.4.3. A ˜D-Type Fiber inside the Insertion. If the singular fiber of ¯B in the loop inside the insertion is of type ˜D5, then relation (3.3.1) should be replaced with
ρβ2β3β−12 ρ−1= β2, ρβ2ρ−1= β3, (3.4.4)
where{β1,β2,β3} is an appropriate canonical basis over the •-vertex w in
Fig-ure 10. Using [10], one hasβ1 = α1α3α1−1,β2 = α1, andβ3 = α−13 α2α3 (in
particular,β1β2β3 = ρ). From (3.4.4) it follows that δ := ρ2β2β3 is a central
element ofπ1; since the projection of δ to the abelianization of π1is the element
−4 of infinite order, one can use Lemma 3.1.3 and study the commutant of the quotientπ1/δ.
The sets of parameters(l, m, n, k) used in the calculation are listed in Table 2. The only nonabelian group in this series is the one corresponding to the set of sin-gularities E6⊕ D5⊕ A5⊕ A3 (no. 16 ); it can be represented as a semidirect
product((Z3× Z3) Z3) Z and is isomorphic to the one described in item (4).
3.5. Hexagon with a Double Loop
Assume that ¯H is a hexagon with a double loop (see Section 2.4), and choose for v the vertex shown in Figure 5, right. We can assume that the singular fibers inside
the insertion are all of type ˜A; see Remark 2.4.1. Then, the braid relations result-ing from the inner pentagon and monogon are
{α1,α3−1α2α3}5= 1, α1α−13 α2α3α1−1= α−13 α−12 α3α2α3. (3.5.1)
In addition, for all curves except no. 37 in Table 1 there is a relation{α1,α2}l= 1,
wherel = 10, 7, 6, 9, or 5 (in the order of appearance in Table 1). For no. 37, one has the commutativity relation [α3,α1α2]= 1 from the ˜D9-type fiber. In all cases,
we can use GAP [13] to conclude that the group is abelian.
The same arguments apply to the set of singularities 2E6⊕ A4⊕ A3 (no. 4
in Table 1) because the corresponding skeleton can be represented as shown in Figure 11(b) (so that one hasl = 4). The resulting group has order 720, and its presentation is
π1= α1,α2,α3| (3.1.1), (3.5.1), {α1,α2}4 = 1. (3.5.2)
This group is isomorphic to (3.3.3); see Remark 2.6.1. 3.6. Genuine Hexagon: Irreducible Curves
Consider one of the four skeletons shown in Figure 7, and take forv any vertex in∂ ¯H. Let v0 = v, v1,..., v5be the vertices in∂ ¯H numbered starting from v in
the clockwise direction (this is the counterclockwise direction in the figures, which represent the complementary hexagonS2\ ¯H ). Mark each vertex similar to v
0 = v
and denote byRithe region of Sk whose boundary contains the edgese1ande2
atvi,i = 0, ..., 5. Let nibe the number of vertices in∂Ri,i = 0, ..., 5; in other words, assume thatRi is anni-gon. Then, in addition to the common relation at infinity (3.1.1), the groupπ1has the relations
{σi
2(α2), α1}ni= 1, i = 0, ..., 5, (3.6.1) resulting from the singular fibers inRi. If Riand ¯H are all the regions but at most one of Sk (which is always the case for irreducible curves; see Figure 7), then (3.1.1) and (3.6.1) form a complete set of relations forπ1. Furthermore, in the
se-quenceR0,..., R5, some of the regions coincide. For each region, it suffices to
consider only one instance in the sequence and ignore the other relations by let-ting the corresponding parametersni equal 0; these relations would follow from the others.
The values of the parameters(n0,..., n5) used in the calculation are listed in
Table 1, and the initial vertexv = v0is shown in Figure 7 in gray. For the set of
singularities E6⊕ A6⊕ A4⊕ A2⊕ A1(no. 42 in Table 1), the resulting group
is abelian. The other three curves are of torus type, so their groups factor to ¯B3.
Here is some additional information on these groups.
(1) E6⊕ A5⊕ 4A2(no. 39): the curve is a sextic of torus type of weight 8 in the
sense of [5]. Henceπ1is much larger than ¯B3; its Alexander module is a
di-rect sum of two copies ofZ[t]/(t2− t + 1). An alternative presentation of this
group is found in [9].
(2) (E6⊕ A8⊕ A2) ⊕ A2⊕ A1(no. 40): much as in Section 3.3.10, one can use
(3) (E6⊕ A5⊕ 2A2) ⊕ A4 (no. 41): the quotientπ1/α15 is a perfect group of
order 7680, whereas ¯B3/σ15 = A5. Hence the natural epimorphism π1 ¯B3
is proper. (The values of the parameters actually used are (n0,..., n5) =
(3, 3, 5, 6, 6, 5).) I do not know whether this group is isomorphic to the one considered in Section 3.3.9.
3.6.2. Remark. In items (2) and (3), if the reference fiber is chosen as shown in Figure 7 then the group has also relation(α1α2)−1α2(α1α2) = α3; see (3.8.1).
Al-though formally this extra relation follows from the others, it simplifies the analysis of the group.
3.7. Genuine Hexagon: Reducible Curves
The approach of Section 3.6 applies to reducible curves ¯B as well (see Figure 8)— provided that the skeleton of ¯B has at most one region or ˜D-type singular fiber strictly inside the complementary hexagonS2 \ ¯H (i.e., to all skeletons except
those shown in Figures 8(f )–( j)). (In the case of a ˜D-type fiber, shown in Fig-ures 8(p) and (q), the regionRicontaining this fiber should be ignored in (3.6.1).) The first two curves (Figures 8(a) and (b)) do not seem to have any extra fea-tures that would facilitate the study of their groups. (Each of these curves splits into an irreducible quintic and a line; henceπ1/[π1,π1]= Z.) The values of the
parameters(n0,..., n5) are listed in Table 2 (assuming that v = v0is the vertex
shown in the figures in gray), and the resulting groups are as follows.
(1) (E6⊕ A5⊕ 2A2) ⊕ A3⊕ A1(no. 20 ): the curve is of torus type, so π1factors
to the braid groupB3; in particular, [π1,π1] is infinite.
(2) E6⊕ A4⊕ 2A3⊕ A2⊕ A1(no. 21 ): the commutant [π1,π1] is perfect (one
can compute the Alexander moduleA = 0 as in Section 3.10); it appears to be infinite, but at present I do not even know whether it is nontrivial.
In the rest of this section we consider the skeletons with one monogonal region insideP1\ H (i.e., those shown in Figures 8(k)–(o)). Choose for the initial
ver-texv = v0the one shown in the figures in gray. Then the monogonal region gives
an extra relation
(α2α1α2)−1α1(α2α1α2) = α3. (3.7.1)
(Strictly speaking, this relation follows from the others, but its presence simpli-fies the calculations. In particular, since each sexticB in question is known to be reducible, it follows that it splits into an irreducible quintic and a line, where the projectionπ1→ π1/[π1,π1]= Z is given by α1,α3 → 1 and α2→ −5.) On the
other hand, relation (3.6.1) implies thatδ := (α1α2)n0commutes withα1andα2;
henceδ is a central element and, using Lemma 3.1.3, one can study the commutant of the quotientπ1/δ.
The parameters(n0,..., n5) are listed in Table 2. The following three sets of
singularities result in nonabelian fundamental groups.
(3) E6⊕ A5⊕ A4⊕ A3⊕ A1(no. 30 ): one has π1= SL(2, F5) Z.
(5) E6⊕ A9⊕ A2⊕ 2A1(no. 34 ): the commutant of π1is a perfect group; it
ap-pears infinite, but I do not know a proof. The commutants ofπ1/α12andπ1/α13
have orders 60 and 51840, respectively.
(In items (3) and (4), the centralizer of [π1,π1] projects to a subgroup of index 2
and 1, respectively, inπ1/[π1,π1]. The groups are isomorphic to those of items
(3) and (2), respectively, in Section 3.4.) The group in item (4) was first computed in [1], where it was also shown that sextics nos. 3 and 33 in Table 2 are Galois conjugate overQ√2.
3.8. Genuine Hexagon: Two Monogons insideS2\ ¯H
In this section, we consider a skeleton with two monogonal regions strictly inside S2\ ¯H (i.e., one of those shown in Figures 8(f)–( j)). Take for v = v
0the
ver-tex shown in the figures in gray. Then, in addition to (3.1.1) and (3.6.1), the group has an extra relation
(α1α2)−1α2(α1α2) = α3 (3.8.1)
resulting from the monogon closest tov. In particular, it follows that the curve splits into an irreducible quartic and an irreducible conic. Furthermore, because δ := (α1α2)n0 commutes withα1andα2, it is a central element and one can use
Lemma 3.1.3 to study the commutant of the quotientπ1/δ.
The parameters(n0,..., n5) are listed in Table 2. For the first three curves, the
groups are abelian. (As a consequence, the curve defined by the skeleton in Fig-ure 8(g), no. 26 in Table 2, is not of torus type.) The other two curves are of torus type; an alternative way to construct these curves and to compute their fundamen-tal groups is found in [8]. (To prove that these curves are of torus type, one can argue that the existence of such curves is shown in [8] and nos. 28, 29 are the only candidates left.)
3.9. Genuine Hexagon with a ˜D-Type Fiber
Consider one of the two skeletons shown in Figure 8(p) or (q), and choose the ini-tial vertexv = v0 so thatR0 is the region containing the ˜D-type fiberF of ¯B.
Then, as before, the defining relations forπ1are (3.1.1) and (3.6.1), with the
con-tribution ofR0ignored in the latter.
However, we do make use of the regionR0 in order to find central elements
inπ1. Let n = n0. Then F is of type ˜Dn+4, and the braid relations aboutF are
α−1
3 αiα3= σ1n+2(αi), i = 1, 2.
As a consequence,
[α3,α1α2]= 1; hence [α3,ρ] = [α1α2,ρ] = 1, (3.9.1)
andδ := α3(α1α2)1+n/2is a central element ofπ1. (Note that n is even in all
cases.) Since [α2α3,ρ4]= 1 (see (3.1.1)), by (3.9.1) one has [α2,ρ4]= 1 and then
[α1,ρ4]= 1. Thus, ρ4= (α2α3)3is also a central element ofπ1, and Lemma 3.1.3
applied twice implies that the commutant [π1,π1] ofπ1is isomorphic to that of
the quotientπ1/δ, ρ4. (It is worth mentioning that n ≥ 2 and so the images of δ
The values of the parameters(n0,..., n5) are listed in Table 2. (Recall that the
initial vertexv0is determined by the position of the ˜D-type fiber, which depends
on the curve.) We can use GAP [13] to conclude that, for the sets of singularities E6⊕ D8⊕ A3⊕ 2A1 and E6⊕ D6⊕ A5⊕ 2A1
(nos. 35 and 38 in Table 2), the groupπ1is abelian whereas for the other two
curves (nos. 36 and 37 ) it has infinite commutant. For a more precise statement see Section 3.10, where we compute the Alexander modules of these and a few other groups.
3.9.2. Remark. The fact that the fundamental groups of the reducible sextics with the sets of singularities E6⊕ D6⊕ 2A3⊕ A1 and E6⊕ D10⊕ 3A1have
infinite commutants can also be explained as follows. Each curve splits into an ir-reducible quarticB4with a type-E6singular point and a pair of lines. One of the
linesB1either is double tangent toB4or has a single point of 4-fold intersection
withB4. Hence, even after patching back in the other line (which corresponds to
letting one of the canonical generators ofπ1equal 1), one obtains a curve with
large fundamental group (which is, respectively,B3orT3,4 = α, β | α3 = β4;
see [3]).
3.10. Other Sextics with Two Linear Components
With the exception of the two curves mentioned in the previous section, all maxi-mizing sextics splitting into an irreducible quartic (with a type-E6singular point)
and two lines have fundamental groups with infinite commutants. In order to prove this statement and make it more precise, we compute the so-called Alexander mod-ules of the groups (see [15]).
3.10.1. Definition. Let G be a group, and let G = [G, G] be its commu-tant. The Alexander module ofG is the abelian group G /[G ,G ] regarded as a Z[G/G ]-module via the conjugation action(a, x) → a−1xa with a ∈ G/G and
x ∈ G /[G ,G ].
Abbreviateπ1 = [π1,π1] and denote the Alexander module ofπ1byA.
The sextics in question are represented by the skeletons shown in Figures 8(c)– (e), (p), and (q), and the defining relations forπ1are (3.1.1) and (3.6.1). (As usual,
if a ˜D-type fiber is present then the corresponding relation in (3.6.1) should be ignored.) The abelianizationπ1/π1 = Z ⊕ Z is generated by the images s, t of
α1,α2, respectively, and the group ringZ[π1/π1 ] can be identified with the ring
+ := Z[s, s−1,t, t−1] of Laurent polynomials ins, t.
Each skeleton in question has a bigonal region not containing a ˜D-type fiber, and we choose the initial vertexv0so that this region isR0. (See the gray vertex
in the figures; note that, for Figures 8(p) and (q), this choice ofv0 differs from
that used in Section 3.9.) Then [α1,α2]= 1 and, using the Reidemeister–Schreier
method (see e.g. [16]), one can see thatA is generated over + by a single ele-menta := α4
1α2α3. The relation at infinity (3.1.1) transforms into (s−4+s−8)a =
Q∞(s)a = 0, where Q∞(s) := (s2− s + 1)(s4− s2+ 1). (3.10.2)
Alternatively, (3.10.2) can be rewritten in the form(s − 1)s(s4+ s + 1)a = −a, which means that the multiplication bys − 1 is invertible in A. For this reason, we cancel the factors − 1 in all other relations.
For an integerm ≥ 0, denote Pm(x) = (xm−1)/(x −1). (In particular, P0≡ 0
andP1≡ 1.) Observe that, for curves with two linear components, all integers ni
in (3.6.1) are even; see Table 2. A relation{σ2i(α2), α1}2r = 1 results in the
fol-lowing relation forA:
(t − 1)Pr(st)Pj(s4)a = 0 if i = 2j is even, or
Pr(s3t)(1 − s4t)Pj(s4) + s4jta = 0 if i = 2j − 1 is odd.
(Recall that we cancel all factorss − 1.) Using the values of (n0,..., n5) listed in
Table 2, one arrives at the following Alexander modules.
(1) E6⊕ A7⊕ A3⊕ 3A1(no. 22 ): A = Z[s]/Q∞(s) and ta = a.
(2) E6⊕ A5⊕ 2A3⊕ 2A1(no. 23 ): A = Z[s]/Q∞(s) and ta = −s−3a.
(3) (E6⊕ 2A5) ⊕ 3A1(no. 24 ): A = Z[s]/(s2− s + 1) and ta = (1 + s−4)a.
(4) E6⊕ D6⊕ 2A3⊕ A1(no. 36 ): A = Z[s]/(s2− s + 1) and ta = −s−3a.
(5) E6⊕ D10⊕ 3A1(no. 37 ): A = Z[s]/Q∞(s) and ta = a.
(Note thatQ∞|(s12− 1); hence s is invertible in Z[s]/Q
∞and one need not
con-sider Laurent polynomials explicitly.)
3.10.3. Remark. In items (1) and (5), one hasn2 = 2; hence (t − 1)a = 0.
In items (2) and (4), one hasn1= 4; hence (1 + s3t)a = 0. In each case, t is a
Laurent polynomial ins and one can represent A as a quotient of the +-module Z[s]/Q∞(with an appropriate action oft); then, in most cases, all extra relations
follow from the relation at infinity (3.10.2).
DenoteQ(s) = s2−s +1 and R(s) = s4−s2+1, so that Q
∞(s) = Q(s)R(s).
In item (4), in addition to (3.10.2) one has the relationQ(s)S1(s)a = 0, where
S1(s) := (s + 1)2. Since R(s) − (s − 1)2S1(s) = s2is an invertible element, the
relation ideal (inZ[s]) is generated by a single element s2− s + 1.
In item (3), the additional relations are
P3(s3t)a = (t − 1)P3(st)a = (1 − s4t + s4)a = 0.
From the last relation, one obtainsta = (1 + s−4)a. Hence, again t is a polyno-mial ins and, substituting t = 1 + s−4into the first two relations, one has
(s2− s + 1)S
1(s)a = (s2− s + 1)S2(s)a = 0,
where
S1(s) := s6+ s5+ 2s2+ 2s + 1, S2(s) := s6+ 2s5+ 2s4+ s + 1.
One can easily check that
s(s3− s − 2)(s4+ s3+ s2− s + 1)R(s) − s2(s4+ s3+ s2− s + 1)S 1(s)
+ (s5+ s4+ s3− s2+ s + 1)S
2(s) = 1.
3.10.4. Remark. In items (1) and (2), the fact that the groups are infinite can be explained similarly to Remark 3.9.2. Each curve splits into an irreducible quar-ticB4 and a pair of lines, one of which is either double tangent toB4 or has a
single point of 4-fold intersection withB4.
4. Perturbations
We fix a maximizing plane sexticB satisfying condition (∗) in the Introduction and consider a perturbationB of B. Throughout this section, ¯B stands for the maximal trigonal curve corresponding toB via Proposition 2.1.1.
4.1. Perturbations of the Type-E6PointP
LetU be a Milnor ball about the distinguished type-E6singular pointP of B. The
groupπ1(U \ B) is generated by three elements β1,β2,β3subject to the relations
β3= (β1β2β3)β2(β1β2β3)−1, β2 = (β1β2β3)β1(β1β2β3)−1. (4.1.1)
According to [11], the inclusion homomorphism π1(U \ B) → π1(P2 \ B) is
given by
β1→ ρα1ρ−1, β2→ α1, β3→ ρ−1α1ρ,
where{α1,α2,α3} is any basis for π1(P2\ B) as in Section 3.1. Note that a
type-E6singularity has an order-3 automorphism inducing the automorphism
β1→ (β1β2)β3(β1β2)−1, β2→ β1, β3→ β2
of the local fundamental group. However, on the image in π1(P2 \ B), this
latter automorphism is inner—namely, it is induced by the conjugation g → ρgρ−1. (For proof, one needs to use the commutativity relation [α
1,(α2α3)3]= 1,
which follows from (3.1.1).) Hence, the local automorphisms atP can be ignored when studying the extra relations inπ1(P2\ B ) resulting from a perturbation
B → B .
The fundamental groupsπ1(U \ B ) of small perturbations B → B are found
in [4]; they are as follows.
(1) E6→ 2A2⊕ A1: one hasπ1(U \ B) = B4, where the additional relations are
[β1,β3]= {β1,β2}3= {β2,β3}3= 1.
(2) E6 → 2A2(as a further perturbation of (1)) and E6 → A5: one hasπ1(U \
B) = B3, where the additional relations areβ1= β3and{β1,β2}3= 1.
(3) All others: one hasπ1(U \ B) = Z, and the relations are β1= β2 = β3.
Note that, in the presence of (4.1.1), the additional relations in items (1) and (2) follow from the first relation; namely, [β1,β3]= 1 and β1= β3, respectively. The
first two perturbations are of torus type (i.e., they preserve torus structures ofB); the other perturbations destroy any torus structure with respect to whichP is an inner singularity.
Combining items (1)–(3) with the inclusion homomorphism, we obtain the fol-lowing extra relations for the perturbed groupπ1(P2\ B ).
(1) E6→ 2A2⊕ A1: the extra relation is [α1,ρ−2α1ρ2]= 1.
(2) E6→ 2A2and E6→ A5: the extra relation is [α1,ρ2]= 1.
(3) All others: the extra relation is [α1,ρ] = 1.
As a consequence, ifP is perturbed as in (3), then one has [α1,α2α3] = 1 and
(3.1.1) becomes
α4
1α2α3= 1. (4.1.2)
In particular,π1is generated byα1andα2(or byα1andα3) in this case.
4.1.3. Lemma. Assume that the distinguished hexagon ¯H of Sk is genuine, that one of the regionsRi(see Section 3.6 ) is a bigon, and that the pointP is perturbed as in item (3). Then the groupπ1(P2\ B ) is abelian.
Proof. TakingRiforR0, one concludes that [α1,α2]= 1. Together with (4.1.2),
this relation implies that the group is abelian.
4.1.4. Remark. Note that almost all skeletons with a genuine hexagon have a bigonal region; the exceptions are Figures 7(a) and (c) and Figure 8( j).
4.2. Perturbations of A-Type Points
Extend the skeleton Sk ofB to a dessin by inserting a ◦-vertex at the middle of each edge, inserting a×-vertexvR at the center of each regionR, and connecting vR to the vertices in the boundary∂R by appropriate edges (see [10]). Let Q be a type-Apsingular point ofB; it is located in a type- ˜Apsingular fiberF of ¯B at the center of a certain(p + 1)-gonal region R of Sk. If Q is perturbed then F splits into singular fibers of types ˜Asi ( ˜A∗0ifsi = 0), i = 1, ..., r, with(si+ 1) = p +1. Geometrically, the 2(p +1)-valent×-vertexvRsplits into several×-vertices of valencies 2(si+ 1).
Assume that the braid relation inπ1(P2\B) resulting from the region R just
de-scribed is of the form{δ1,δ2}p+1= 1, where δ1,δ2are certain words inα1,α2,α3
(cf. (3.3.2), (3.4.1), (3.6.1)). In other words,δ1andδ2are the first two elements of
a canonical basis over a vertexu ∈ ∂R defined by a marking at u (see [10]) with respect to whiche1ande2belong to∂R.
4.2.1. Lemma. In the notation just introduced, the relation for the new group π1(P2\ B ) resulting from R is {δ1,δ2}s = 1, where s = g.c.d.(si+1), i = 1, ..., r.
Proof. The statement follows from the description of the braid monodromy found in [10] and from (3.1.4).
4.2.2. Corollary. IfQ is perturbed then the relation {δ1,δ2}p+1= 1 changes
to the relation{δ1,δ2}s= 1, where s < p +1 is a divisor of (p +1). In particular,
one hass = 1 if (p + 1) is a prime.
4.2.3. Corollary. IfQ is a point of intersection of two components of B (and hencep is odd ) and if the perturbation is to be irreducible in a Milnor ball about Q, then s in Corollary 4.2.2 must be an odd divisor of p + 1.
4.2.4. Corollary. IfB is of torus type and Q is an inner singularity (hence p + 1 = 0 mod 3) and if the torus structure is to be destroyed, then s in Corol-lary 4.2.2 must not be divisible by 3.
4.3. Perturbations of D-Type Points
LetQ be a type-Dp,p ≥ 5, singular point of B; it is located in a type- ˜Dpsingular fiberF of ¯B at the center of a (p − 4)-gonal region R of Sk.
4.3.1. Lemma. If a perturbationB → B is irreducible in a Milnor ballUQ
aboutQ, then the group π1(P2\ B ) is abelian.
Proof. Under the assumptions, the groupπ1(UQ\ B ) is abelian (see [12]). On
the other hand, the inclusion homomorphismπ1(UQ\ B ) → π1(P2\ B ) is onto,
sinceQ is a triple point of a trigonal curve.
Letδ1 andδ2 be the first two elements of a canonical basis over a vertexu ∈ ∂R
defined by a marking atu (see [10]) with respect to which e1ande2belong to∂R.
(In other words, ifR contained an ˜A-type fiber then the resulting braid relation would be{δ1,δ2}p−4= 1; cf. (3.3.2), (3.4.1), (3.6.1).)
4.3.2. Lemma. Assume that a proper perturbationB → B is still reducible in a Milnor ballUQaboutQ. Then, with notation as before, the new group π1(P2\ B)
has an extra relation{δ1,δ2}s = 1 for some integer s, 1 ≤ s ≤ p − 2. If B ∩ UQ
has three components(p is even) but B ∩ UQhas two components, thens is odd. Proof. The statement follows from the computation of the fundamental group π1(UQ\ B ) found in [12].
4.4. Proof of Theorem 1.2.3
We skip sextics of torus type with two or more type-E6singular points (nos. 1, 2,
6, and 7 in Table 1; they are considered in detail in [4]) as well as the maximizing sextic of weight 8 (no. 39 in Table 1; considered in [9]).
All other perturbations with nonabelian groupπ1:= π1(P2\ B) can be handled
on a case-by-case basis, using GAP [13] and modifying the presentation forπ1
found in Section 3 according to Section 4.1, Corollaries 4.2.2–4.2.4, and Lem-mas 4.3.1 and 4.3.2. (Recall that, by [9, Prop. 5.1.1], all singular points ofB can be perturbed arbitrarily and independently.) A few details are given in Sections 4.4.1– 4.4.5. In some cases, the same approach shows that any proper perturbation, in-cluding reducible ones, that is not of torus type has abelian fundamental group.
• no. 1 : All perturbations not of torus type are abelian.
• no. 3 : All proper perturbations are abelian (see Section 4.4.3). • nos. 6 , 15 , 16 : All proper perturbations are abelian.
• no. 20 : All perturbations not of torus type are abelian (see Section 4.4.4). • no. 24 : All perturbations not of torus type are abelian (see Section 4.4.5). • nos. 28 , 29 : All perturbations not of torus type are abelian.
